The following are expanded lecture notes for the course of eight one hour lectures given by the second author at the 2014 summer school Asymptotic Analysis in General Relativity held in Grenoble by the Institut Fourier. The first four lectures deal with conformal geometry and the conformal tractor calculus, taking as primary motivation the search for conformally invariant tensors and diffrerential operators. The final four lectures apply the conformal tractor calculus to the study of conformally compactified geometries, motivated by the conformal treatment of infinity in general relativity.

Past research in integrated three-dimensional (3D) craniofacial mapping at the Craniofacial Research Instrumentation Laboratory (CRIL) of the University of the Pacific is summarized in narrative form. The advantages and limitations of recent commercial developments in the application of cone beam geometry volumetric X-ray scanners in dentistry and surface digital mapping of study casts are discussed. The rationale for methods currently in development at CRIL for merging longitudinal information from existing 3D study casts and two-dimensional lateral X-ray cephalograms in studies of orthodontic treatment outcome is presented.

Objectives: To quantify reliability of three-dimensional skeletal landmarks and a comprehensive set of dental landmarks in cone-beam computed tomography (CBCT) and to determine the shapes of envelope of error. Materials and Methods: Three judges located 31 skeletal landmarks and 60 dental landmarks on the pre- and posttreatment CBCT images of 22 patients. Landmark error was determined by calculating the distance of deviation of landmark locations around their average. Standard deviation and mean radial spherical error were calculated. Scatterplots were constructed to characterize envelope of error. Results: The midline landmarks of the cranial base were highly reliable. Bilateral skeletal landmarks tended to have larger error than midline landmarks. Among the nonconventional landmarks, fronto-zygomatic suture, condyle, and mental foramen showed relatively high reliability. However, foramen spinosum and temporal fossa showed larger errors. Gonion was the least reliable landmark. Most dental landmarks were located more reliably than skeletal landmarks. The highest reliability was found at incisal edges. Mesiobuccal cusp of first molars also showed high reliability. Conclusions: There were differences in the size and shape of the distributions of errors of different landmarks. Most landmarks showed elongated envelopes. Bilateral structures tended to show greater errors than midline structures. Most dental landmarks were more reliable than skeletal landmarks.

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